{
  "counts": {
    "adjacent": 15,
    "core": 2,
    "errors": 0,
    "negative": 73,
    "total": 90
  },
  "date": "2026-09-17",
  "errors": [],
  "fresh_content_days": 21,
  "generated_at": "2026-09-17T19:12:17Z",
  "items": [
    {
      "age_days": 2,
      "arxiv_id": "2609.16991",
      "authors": [
        "Xin Quan",
        "Reto Gubelmann",
        "André Freitas"
      ],
      "content_date": "2026-09-15",
      "freshness": "fresh",
      "id": "arxiv:2609.16991",
      "kind": "paper",
      "label": "core",
      "matched_signals": [
        "autoformalization",
        "verifier_guided_reasoning"
      ],
      "published": "2026-09-15",
      "score": 9.5,
      "source": "arxiv-ai4math-core",
      "summary": "Natural language arguments are compelling before they are formally explicit. A premise supports a claim through defeasible warrants, background commitments, and exception conditions that the text leaves implicit. However, formal verification requires the opposite. Making such arguments machine-checkable requires constructing the missing commitments, not only translating given sentences into logic. Construction, however, carries a risk that translation does not: a system free to add premises can make any claim provable, and a formally valid proof may assert the claim outright, prove it without the original premise, or establish more than the claim itself. We address this problem by formulating autoformalization for argumentative material inference as guard completion, in which non-monotonic material support is turned into monotonic formal inference relative to an explicitly constructed guard set. A completion is accepted only when its proof both passes the theorem prover and survives contrastive tests of premise dependence and claim selectivity. We implement this formulation in GUARD, a neuro-symbolic framework in which LLMs construct and formalize candidate guards, Isabelle/HOL verifies the resulting theories and returns step-level feedback for iterative refinement, and the system abstains when no faithful completion can be reached. Our empirical results on Debatepedia and ARCT using different LLMs demonstrate that GUARD yields significant improvements in verified-faithful (+35.3, +32.9 points) and substantial reductions in leakage (-25.9, -21.9 points) over the state-of-the-art LLM-driven theorem proving approach. Moreover, we show that the symbolic soft critique and the explicit assumption layer account for most of these gains, with the soft critique also improving the initial validity of the elicited context and reducing the number of iterations required for successful verification.",
      "title": "Autoformalizing Argumentative Material Inferences",
      "updated": "2026-09-15",
      "url": "https://arxiv.org/abs/2609.16991"
    },
    {
      "age_days": 4,
      "arxiv_id": "2609.14808",
      "authors": [
        "Andre Panossian"
      ],
      "content_date": "2026-09-13",
      "freshness": "fresh",
      "id": "arxiv:2609.14808",
      "kind": "paper",
      "label": "core",
      "matched_signals": [
        "autoformalization",
        "verifier_guided_reasoning"
      ],
      "published": "2026-09-13",
      "score": 6.5,
      "source": "arxiv-ai4math-core",
      "summary": "Scientific autoformalization turns verbal accounts into executable mathematics, but executable code does not settle which model has been constructed. We examine two sources of structural uncertainty: the formalizer that generates a response law, and the recurrence that turns that law into trajectories. In secondary analyses of an openly archived crossed experiment, we studied 320 response maps generated by two pinned language-model formalizers from five engineered cognitive accounts within one sparse quadratic grammar and 16 randomized blocks. With whole blocks held out, source-account identity was recovered at 78.8% accuracy (chance 20.0%) and formalizer identity at 96.3% (chance 50.0%; both p < 0.001). Program size was the stronger single feature family; a pre-specified exploratory comparison found no stable source-account predictive gain from local geometry beyond size. Holding every response map fixed, we then evaluated five recurrence families spanning 33 configurations and 1,013,760 finite-horizon trajectories. Added feedback, projection and leak produced sharply different outcome distributions. The consequential distinction was which comparisons survived: median cross-recurrence rank concordance was 0.73 for endpoint magnitude but 0.05 for settling, among the configuration pairs with defined rankings. Thus a common mathematical language did not erase translation provenance, and robust ordering under one observable did not transfer to another. Scientific autoformalization is usefully studied as model-space construction: the generated ensemble and its dynamical embedding are both part of the specification supporting a scientific claim.",
      "title": "Transformed in Translation: Two-Stage Structural Uncertainty in LLM-Based Scientific Autoformalization",
      "updated": "2026-09-13",
      "url": "https://arxiv.org/abs/2609.14808"
    },
    {
      "age_days": 1,
      "arxiv_id": "2609.19126",
      "authors": [
        "Teng Zhang"
      ],
      "content_date": "2026-09-16",
      "freshness": "fresh",
      "id": "arxiv:2609.19126",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "general_ai_math_reasoning",
        "lean_formal_proving_agents"
      ],
      "published": "2026-09-16",
      "score": 4.9,
      "source": "arxiv-ai4math-core",
      "summary": "Very recently, Lech Mazur proved the celebrated Sendov conjecture, and Terence Tao subsequently distilled the main ideas of the proof in a blog post. In this paper, we establish a quantitative strengthening of Sendov's conjecture, namely the quadratic Tang--Zhang inequality. Let $p$ be a polynomial of degree $n\\ge2$ whose zeros lie in the closed unit disk, and let $ζ_1,\\ldots,ζ_{n-1}$ denote its critical points, counted with multiplicity. We prove that, for every zero $a$ of $p$, $$ \\sum_{j=1}^{n-1}\\frac{1}{|a-ζ_j|^2}\\ge n-1. $$ Moreover, equality holds if and only if $p(z)=c(z^n-ω)$ for some $c\\in\\mathbb C\\setminus\\{0\\}$ and $|ω|=1$. We also provide a Lean 4 formalization of the main results.",
      "title": "Beyond Sendov's conjecture: the quadratic Tang--Zhang inequality",
      "updated": "2026-09-16",
      "url": "https://arxiv.org/abs/2609.19126"
    },
    {
      "age_days": 4,
      "arxiv_id": "2609.14351",
      "authors": [
        "Yinjie Li"
      ],
      "content_date": "2026-09-13",
      "freshness": "fresh",
      "id": "arxiv:2609.14351",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "general_ai_math_reasoning",
        "lean_formal_proving_agents"
      ],
      "published": "2026-09-13",
      "score": 4.9,
      "source": "arxiv-ai4math-core",
      "summary": "We prove the Colomo-Pronko conjecture for alternating sign matrices with a prescribed square of zeros at a corner, for all matrix sizes and freezing parameters. A known multiple-integral formula for the frozen-corner count yields determinant representations built from fixed polynomial kernels. We relate these kernels to the conjectured determinant through an inverse identity for the commutator of a signed Pascal matrix with reversal. In odd dimension, the comparison uses the one-dimensional nullspace and projection along it to eliminate the central coordinate. Combined with the asymptotic analysis of Colomo and Pronko, our result removes the conjectural assumption from their GUE Tracy-Widom fluctuation theorem for the intersection of the frozen boundary with the main diagonal in uniformly random alternating sign matrices. The finite-dimensional algebraic core of the proof has been formalized in Lean 4.",
      "title": "The Colomo-Pronko conjecture for frozen-corner alternating sign matrices",
      "updated": "2026-09-13",
      "url": "https://arxiv.org/abs/2609.14351"
    },
    {
      "age_days": 1,
      "arxiv_id": "2609.18261",
      "authors": [
        "Pierre Senellart",
        "Anton Gnatenko"
      ],
      "content_date": "2026-09-16",
      "freshness": "fresh",
      "id": "arxiv:2609.18261",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "mathlib_retrieval"
      ],
      "published": "2026-09-16",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "We show that descriptive complexity can serve as a foundation for formalizing computational complexity results in a proof assistant, by constructing a Lean library centered around the following concepts: decision problems are isomorphism-invariant predicates on finite structures; complexity classes are defined by their logical characterization; membership is shown by definability witnesses; hardness is shown by first-order reductions from a known hard problem. We also establish bridges to traditional machine models such as (non)deterministic Turing machines. The library proves 73 completeness results, on 68 problems or problem families, over 14 different classes; relations between the classes established inside the logic and not by machine simulation, among them NL = coNL and the Abiteboul-Vianu theorem; and unconditional lower bounds, among them $\\mathrm{FO}(\\leq) \\subsetneq \\mathrm{FO}(\\leq, \\mathrm{TC})$ and the failure of order-free FO(IFP) to capture PTIME.",
      "title": "Descriptive Complexity in Lean: Completeness by First-Order Reductions",
      "updated": "2026-09-16",
      "url": "https://arxiv.org/abs/2609.18261"
    },
    {
      "age_days": 2,
      "arxiv_id": "2609.17027",
      "authors": [
        "Junjie Liao"
      ],
      "content_date": "2026-09-15",
      "freshness": "fresh",
      "id": "arxiv:2609.17027",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-09-15",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "Let $F=F(X)$ be a free group of finite rank, with palindromic length taken with respect to the fixed basis $X$. We embed $F$ as the index-two subgroup of the universal Coxeter group $W=F\\rtimes_θ\\langle t\\mid t^2=1\\rangle$, where $θ(x)=x^{-1}$ for $x\\in X$, and prove $\\mathrm{pl}(g)=\\min{\\ell_T(g),\\ell_T(gt)}$. Dyer's deletion theorem then identifies reflection length with the minimum number of unmatched positions in a noncrossing equal-label partial matching on a reduced Coxeter word. This gives an $O(n^3)$-time, $O(n^2)$-space algorithm for palindromic length, together with recovery of an optimal palindromic factorization. The matching model also gives a structural characterization. For every ordered full binary tree with $k$ leaves we define a literal word template whose leaves are palindromes and whose internal vertices carry arbitrary words. A reduced word $w$ represents an element of palindromic length at most $k$ if and only if $w$ is a literal instance of one of these templates. Hence the $C_{k-1}$ ordered binary-tree shapes give a complete finite family for each fixed $k$. For $k=4$ the five templates are exactly the five forms proposed by Frid, proving the completeness of that list. A companion Lean 4 development verifies the four-palindrome classification end to end for every finite rank, including the ordinary reduced-word formulation and the literal five-form conclusion.",
      "title": "Palindromic Length in Free Groups: Reflections, Noncrossing Matchings, and Catalan Forms",
      "updated": "2026-09-15",
      "url": "https://arxiv.org/abs/2609.17027"
    },
    {
      "age_days": 2,
      "arxiv_id": "2609.17283",
      "authors": [
        "Antonio Acuaviva"
      ],
      "content_date": "2026-09-15",
      "freshness": "fresh",
      "id": "arxiv:2609.17283",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-09-15",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "We construct two complemented subspaces of $L_1[0,1]$. The first has the Schur property but fails the Radon--Nikodým property. The second contains a copy of $\\ell_2$ but no copy of $L_1[0,1]$. Neither space is isomorphic to a Banach lattice. This gives a negative answer to the complemented-subspace question of Lindenstrauss and Rosenthal. A Lean 4 formalisation of the main results accompanies the paper.",
      "title": "On complemented subspaces of $L_1[0,1]$",
      "updated": "2026-09-15",
      "url": "https://arxiv.org/abs/2609.17283"
    },
    {
      "age_days": 2,
      "arxiv_id": "2609.16801",
      "authors": [
        "Jacob Bedrossian"
      ],
      "content_date": "2026-09-15",
      "freshness": "fresh",
      "id": "arxiv:2609.16801",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-09-15",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "We present a formalization in Lean 4 of Mouhot and Villani's theorem on nonlinear Landau damping in $\\mathbb T^d$ in all Gevrey regularity indices $s > 1/3$ for small backgrounds.",
      "title": "Formalization of Landau damping in the Vlasov--Poisson equations in Lean",
      "updated": "2026-09-15",
      "url": "https://arxiv.org/abs/2609.16801"
    },
    {
      "age_days": 2,
      "arxiv_id": "2609.16485",
      "authors": [
        "Alex Borisevich"
      ],
      "content_date": "2026-09-15",
      "freshness": "fresh",
      "id": "arxiv:2609.16485",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-09-15",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "We develop a certified continuation framework for equilibrium computation and for training deep equilibrium networks (DEQs), with training formulated as interpolation to accuracy $2^{-b}$. For inference, compact input homotopy selects a unique branch from a supplied start root, and a rounded Newton tracker follows it under certified boundary, conditioning, derivative, and tube-radius bounds. For training, we augment local-plus-low-rank recurrence with programmable dormant bilinear rank-one channels. Loaded Tikhonov solves diagnose a failed interpolation pass without spectral decomposition; an output-preserving repair aligned with the pass residual supplies the required direction. Training requires certified gate realization and column stability on each pass region, well-posed inference, and finite-update error budgets. With polynomial geometric, encoding, precision, and complete backend budgets, both certified inference and training have bit cost $O(\\operatorname{poly}(L+b))$, where $L$ is the encoded instance length. The trainer uses $O(b+\\ell)$ passes and reserve channels from an initial residual bounded by $2^\\ell$. These guarantees concern a certified promise class. Lean 4 verifies the quantitative core and concrete inference backend; numerical comparisons illustrate the loaded mechanism.",
      "title": "Certified Inference and Training for Deep Equilibrium Networks: A Continuation Framework with Polynomial Complexity Guarantees",
      "updated": "2026-09-15",
      "url": "https://arxiv.org/abs/2609.16485"
    },
    {
      "age_days": 3,
      "arxiv_id": "2609.15642",
      "authors": [
        "Henning Ulfarsson"
      ],
      "content_date": "2026-09-14",
      "freshness": "fresh",
      "id": "arxiv:2609.15642",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-09-14",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "We give an exact algorithm counting the permutations that avoid a fixed pattern from the following family: the direct sum of an increasing pattern and the pattern 231. The first members of the family are 1342 and 12453. For each member, the algorithm computes the number of avoiding permutations of every length up to a given bound using polynomially many arithmetic operations and polynomially many stored integers, with degrees that grow linearly in the length of the pattern. We first obtain an exact recurrence by reading a permutation from left to right and recording, at each step, the constraints that the letters read so far impose on those still unread. Its state space grows exponentially, so evaluating it directly takes exponential time. We then show that part of the state is protected: later steps carry it along unchanged and do not depend on it. Factoring the protected part out turns the recurrence into a dynamic program with polynomially many stored transfer entries, and this gives the polynomial bounds for every member of the family. For the pattern 12453, a translation symmetry sharpens the bounds to degree seven for the operations and degree four for the storage. Separately written implementations and exact Chinese-remainder certification determine the number of 12453-avoiding permutations of every length up to 150. The previously published series reached length 38. The same tables also generate uniformly random avoiders in polynomial time. We illustrate this with a heatmap of one million 12453-avoiding permutations of length 300 sampled with floating-point tables. The counting recurrences for 1342 and 12453 are verified in the Lean 4 proof assistant.",
      "title": "Protected tails and polynomial-time enumeration of permutations avoiding a direct sum of an increasing pattern and 231",
      "updated": "2026-09-14",
      "url": "https://arxiv.org/abs/2609.15642"
    },
    {
      "age_days": 3,
      "arxiv_id": "2609.14879",
      "authors": [
        "Aaron Gregory"
      ],
      "content_date": "2026-09-14",
      "freshness": "fresh",
      "id": "arxiv:2609.14879",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-09-14",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "A stencil computation repeatedly updates every cell of a grid from its neighbours' values at the previous timestep. Simulating T steps on N cells directly costs Theta(NT), and a line of work beginning with Ahmad et al. reduces this by composing many timesteps into one linear operator and applying it with a Fast Fourier Transform. That technique needs to know which cells will still obey the same operator when the composed step ends, and in a free-boundary problem they do not: the region governed by a given rule is determined by the solution and moves as it evolves. We study one spatial dimension, a three-point stencil with time-varying coefficients, and a computed region that is a single interval whose two endpoints move by arbitrary amounts at every step, revealed online. Let B be the horizon plus the total variation of the boundary trajectory. We give a schedule whose work is O((B+N) log T log(N+B)) and whose span is O(T log T log(N+B)), and we prove that the values it computes are exact. The best existing bound for a region that moves requires its boundary to travel at most one cell per timestep. We drop that requirement and lose nothing by it: a boundary obeying it has B <= 3T, so our bound stays near-linear on every trajectory the earlier result covers. Elsewhere, B grows only by the distance the boundary actually travels -- one jump of width N costs T + 2N. The reason total variation suffices is that everything the two endpoints touch over a time window of any length lies in two intervals, one per endpoint. This cannot be relaxed: with p regions the bound degrades by a factor p, and at p = sqrt(T) there is an instance on which the work is Theta(T^{3/2}) while B + N = Theta(T). All results are machine-checked in Lean 4, apart from the classical convolution bound, which is imported as an interface.",
      "title": "Fast Stencil Computations on a Single Arbitrarily Moving Interval",
      "updated": "2026-09-14",
      "url": "https://arxiv.org/abs/2609.14879"
    },
    {
      "age_days": 3,
      "arxiv_id": "2609.14912",
      "authors": [
        "Celio Boulay",
        "Alexander Chai",
        "Anthony Chang",
        "Thomas Moulin"
      ],
      "content_date": "2026-09-14",
      "freshness": "fresh",
      "id": "arxiv:2609.14912",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-09-14",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "The mathematics of Origami have been well studied and shown to develop several interesting results. We use Lean 4 tactics and build on Mathlib to redefine the 7 Huzita operations as theorems instead of axioms and prove their existence. We develop proofs for important origami constructions (such as trisecting an angle), implement origami-constructible numbers and prove the associated Cardano's formula, and formalize Haga's theorem. A Crease Pattern Inspector explores physical folding by providing a full pipeline to create and visualize models constrained by the Huzita formalism. The Lean codebase brings 100+ theorems and lemmas.",
      "title": "A Lean Paper About Paper: A Formal Framework for Origami",
      "updated": "2026-09-14",
      "url": "https://arxiv.org/abs/2609.14912"
    },
    {
      "age_days": 1,
      "authors": [
        "Jiedong Jiang"
      ],
      "content_date": "2026-09-16",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:4075d7326a09",
      "kind": "github_update",
      "label": "adjacent",
      "matched_signals": [
        "seed_author:Jiedong Jiang"
      ],
      "published": "2026-09-16",
      "repo": "leanprover-community/mathlib4",
      "score": 2.0,
      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: fix(Algebra/MonoidWithZeroHom): normalize to `.toMonoidWithZeroHom` (#43728)",
      "updated": "2026-09-16",
      "url": "https://github.com/leanprover-community/mathlib4/commit/4075d7326a0909dc38fa4f499d12335b68b09c73"
    },
    {
      "age_days": 1,
      "arxiv_id": "2609.18879",
      "authors": [
        "Ken Ono",
        "Ashvin Swaminathan"
      ],
      "content_date": "2026-09-16",
      "freshness": "fresh",
      "id": "arxiv:2609.18879",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "general_ai_math_reasoning"
      ],
      "published": "2026-09-16",
      "score": 1.9,
      "source": "arxiv-ai4math-core",
      "summary": "Browning and Sawin conjectured that random hypersurfaces with sign coefficients are smooth with probability tending to one as the degree grows. We prove this conjecture and obtain a quantitative bound. For each $n\\geq1$, a degree $d$ form in $n+1$ variables, with independent uniform coefficients in $\\{-1,1\\}$, defines a singular complex hypersurface with probability $O_n(d^{-1/2})$. The positive-dimensional singular loci occur with exponentially small probability. For $n\\geq3$, the same exponential bound holds for failure of absolute irreducibility. These results have been formalized in Lean by AxiomProver assuming existing literature.",
      "title": "On a conjecture of Browning and Sawin on random hypersurfaces with sign coefficients",
      "updated": "2026-09-16",
      "url": "https://arxiv.org/abs/2609.18879"
    },
    {
      "age_days": 1,
      "arxiv_id": "2609.18913",
      "authors": [
        "Tarun Kathuria"
      ],
      "content_date": "2026-09-16",
      "freshness": "fresh",
      "id": "arxiv:2609.18913",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "general_ai_math_reasoning"
      ],
      "published": "2026-09-16",
      "score": 1.9,
      "source": "arxiv-ai4math-core",
      "summary": "\\cite{mss2015} proved Weaver's discrepancy result existentially, resolving the Kadison--Singer conjecture . Finding such signs efficiently for general inputs remained an open algorithmic question. In the real-arithmetic model, we give a deterministic algorithm running in polynomial time with discrepancy at most $35\\sqrt\\varepsilon$. The algorithm walks from the origin of the hypercube to a vertex, fixing coordinates as they hit a face. Its potential measures a soft spectral edge of the discrepancy matrix perturbed by an operator-valued free semicircular element. The perturbation's covariance vanishes as the coefficients reach their endpoints. Inspired by the free interpolation approach of Bandeira, Boedihardjo, and van Handel \\cite{bbvh2023}, we combine Lehner's variational formula \\cite{lehner1999} with spectral Tsallis--$1/2$ regularization used in \\cite{allenZhuLiaoOrecchia2015} and \\cite{pesentivladu2026}. The resulting potential has a finite-dimensional SDP formulation, allowing the discrepancy and remaining covariance to be analyzed together. We analyze the optimizer's stability through the linearized Karush--Kuhn--Tucker (KKT) system of a regularized min--max problem, whose stationarity equations are related to the matrix Dyson equation \\cite{erdos2019}. This gives the movement rule: either a coordinate can move toward its nearer endpoint at small spectral cost, or a low-curvature direction orthogonal to the current coefficient vector allows further progress. Choosing the better sign of this direction controls discrepancy while increasing the squared distance from the origin. Upcoming work \\cite{kathuria2026higherRank} will address higher-rank Kadison-Singer and spectrally thin trees. Lean formalizations of our main discrepancy theorems have been completed and will be released shortly.",
      "title": "A Walk From Free Probability to Matrix Discrepancy II: Weaver's Problem and the Kadison-Singer Conjecture",
      "updated": "2026-09-16",
      "url": "https://arxiv.org/abs/2609.18913"
    },
    {
      "age_days": 1,
      "arxiv_id": "2609.18914",
      "authors": [
        "Tarun Kathuria"
      ],
      "content_date": "2026-09-16",
      "freshness": "fresh",
      "id": "arxiv:2609.18914",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "general_ai_math_reasoning"
      ],
      "published": "2026-09-16",
      "score": 1.9,
      "source": "arxiv-ai4math-core",
      "summary": "The Matrix Spencer conjecture asks whether any $n$ real symmetric matrices A_1,...,A_n \\in \\mathbb{R}^{m \\times m} of operator norm at most one admit a signing $x\\in\\{-1,1\\}^n$ such that the operator norm of the signed sum is at most O(\\sqrt{n \\log(2m/n)}) We give a randomized algorithm establishing this bound with polynomial runtime in the real-arithmetic model. We first prove the $O(\\sqrt n)$ bound for $m\\le n$, resolving the square case, and then obtain the rectangular bound by changing the regularizer. As in earlier algorithmic discrepancy methods \\cite{lovettmeka2012,bansalLaddhaVempala2022,pesentivladu2026}, we run a covariance-controlled random walk from the origin of the hypercube, rounding coordinates near its faces and keeping them fixed. Our potential measures a soft spectral edge of the evolving discrepancy matrix perturbed by an operator-valued free semicircular element. Inspired by the free interpolation approach of \\cite{bbvh2023}, we combine Lehner's variational formula for the free edge \\cite{lehner1999} with spectral Tsallis regularization \\cite{allenZhuLiaoOrecchia2015,pesentivladu2026}. This puts the discrepancy and remaining covariance in a single smooth optimization problem. The potential has a finite-dimensional semidefinite formulation. Stability of its optimizer, governed by equations related to the matrix Dyson equation \\cite{erdos2019}, lets us find a large subspace in which to move while controlling discrepancy. The square case uses the Tsallis--$1/2$ regularizer; the rectangular case uses a suitable generalized Tsallis power regularizer. Our companion paper \\cite{kathuria2026ks} applies these ideas to give an algorithmic proof of Weaver's discrepancy theorem, whose existence proof by [MSS15] resolved the Kadison--Singer conjecture \\cite{mss2015}.Lean formalizations of our main discrepancy theorems have been completed and will be released shortly.",
      "title": "A Walk From Free Probability to Matrix Discrepancy I: Matrix Spencer",
      "updated": "2026-09-16",
      "url": "https://arxiv.org/abs/2609.18914"
    },
    {
      "age_days": 3,
      "arxiv_id": "2609.15757",
      "authors": [
        "Julius A. Zeiss"
      ],
      "content_date": "2026-09-14",
      "freshness": "fresh",
      "id": "arxiv:2609.15757",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
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      "published": "2026-09-14",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: fix: remove lossy syntax separator array coercions (#15020)",
      "updated": "2026-09-14",
      "url": "https://github.com/leanprover/lean4/commit/088be5f36f2ec96da3279f0e56f164dc6a2e7ce9"
    },
    {
      "age_days": 3,
      "authors": [
        "Marc Huisinga"
      ],
      "content_date": "2026-09-14",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:9632ea4c76f8",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-14",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: fix: keep the space before a `hygieneInfo` antiquotation (#15025)",
      "updated": "2026-09-14",
      "url": "https://github.com/leanprover/lean4/commit/9632ea4c76f83a21119f89a14871b3236d4d51d9"
    },
    {
      "age_days": 3,
      "authors": [
        "Henrik Böving"
      ],
      "content_date": "2026-09-14",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:dc34e5f5cf9c",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-14",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: feat: support loading lake check and lake comparator input directly from export files (#15157)",
      "updated": "2026-09-14",
      "url": "https://github.com/leanprover/lean4/commit/dc34e5f5cf9c42dd783471b525abbe66c0194270"
    },
    {
      "age_days": 3,
      "authors": [
        "Gaëtan Serré"
      ],
      "content_date": "2026-09-14",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:d3a0781b9001",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-14",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: feat: add `Float.fma` and `Float32.fma` with logical model (#15024)",
      "updated": "2026-09-14",
      "url": "https://github.com/leanprover/lean4/commit/d3a0781b9001b66763f1e61c400cbc5f49abb5f5"
    },
    {
      "age_days": 3,
      "authors": [
        "Henrik Böving"
      ],
      "content_date": "2026-09-14",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:a2233a321503",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-14",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: feat: add --inadvisably-no-sandbox to comparator (#15156)",
      "updated": "2026-09-14",
      "url": "https://github.com/leanprover/lean4/commit/a2233a3215034a623e36ffda47bf7ed2458cca29"
    },
    {
      "age_days": 3,
      "authors": [
        "Garmelon"
      ],
      "content_date": "2026-09-14",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:e20f62782c5e",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-14",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: chore: add downstream labels via comment (#15155)",
      "updated": "2026-09-14",
      "url": "https://github.com/leanprover/lean4/commit/e20f62782c5e3293519c1db2c4b1e0a2e27b21a3"
    },
    {
      "age_days": 1,
      "arxiv_id": "2609.18722",
      "authors": [
        "Shuxing Yang",
        "Rui Zhao",
        "Junyao Wu",
        "Yize Wang",
        "Wenhao Li",
        "Fujia Chen",
        "Taowen Deng",
        "Shenzhan Hong",
        "Yaqi Li",
        "Zichen Li",
        "Jincheng Mi",
        "Yuang Pan",
        "Kaihao Zhu",
        "Junjie Yang",
        "Hongsheng Chen",
        "Yihao Yang"
      ],
      "content_date": "2026-09-16",
      "freshness": "fresh",
      "id": "arxiv:2609.18722",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-16",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "We prove that the tensor rank of $3\\times3$ matrix multiplication over $\\mathbb F_2$ is at least $21$. The structural proof, independently developed by Qiushi Engine, converts occupation constraints on a single tensor factor into algebraic relations coupling all three factors. Certified quotient-rank bounds and finite geometry force any hypothetical $20$-term decomposition to have first-factor matrix-rank profile $(16,1,3)$. The ranks of the corresponding split-flattened summands therefore sum to $27$, exactly the rank of the full split flattening. Equality in rank subadditivity forces their images to form a direct sum; normalization by the inverse flattening then makes the summands pairwise annihilating idempotents. An explicit product identity for matrix multiplication implies that at most one first factor can be invertible, contradicting the three forced by the profile. The same obstruction constrains $22$-term decompositions attaining the split-rank bound. The complete proof, including the finite quotient bounds, is formalized in Lean. The accompanying research trajectory records Qiushi Engine's long-horizon autonomous research, from numerical experiments and quotient constructions to the structural proof.",
      "title": "A Structural Proof of the Lower Bound 21 for $3\\times3$ Matrix Multiplication over $\\mathbb F_2$",
      "updated": "2026-09-16",
      "url": "https://arxiv.org/abs/2609.18722"
    },
    {
      "age_days": 2,
      "arxiv_id": "2609.17643",
      "authors": [
        "Saurabh Verma",
        "Manish Yadav",
        "Archana Dixit",
        "Anirudh Pradhan",
        "M. S. Barak"
      ],
      "content_date": "2026-09-15",
      "freshness": "fresh",
      "id": "arxiv:2609.17643",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-15",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Late-time cosmic acceleration is conventionally ascribed to a cosmological constant, though $Λ$CDM continues to face several theoretical difficulties that keep alternative gravity models under active consideration. This work examines the Hu-Sawicki $f(R)$ model in a spatially non-flat background, with the modified-gravity parameter $b$ and the curvature density $Ω_k$ both treated as free parameters, constrained using DESI-DR2 BAO, BBN, and four Type Ia supernova compilations -- PantheonPlus, PantheonPlus+SH0ES, Union3, and DESY5yr. Across all four combinations, $H_0$ and $Ω_m$ stay close to their $Λ$CDM values, with a noticeable shift in $H_0$ appearing only for the SH0ES-calibrated dataset. The parameter $b$ departs from zero at better than $2σ$ in three of the four fits, most prominently for DESY5yr, whereas the SH0ES-calibrated combination alone prefers $b<0$. A strong positive correlation among $b$, $H_0$, and $Ω_k$ underlies this shift, suggesting that curvature signatures obtained under $Λ$CDM can be reabsorbed into the modified-gravity sector once this additional freedom is allowed. Statistical model comparison via AIC and BIC gives a mixed picture: AIC leans toward the Hu-Sawicki model in three of the four combinations, while BIC's heavier penalty on the extra parameter favors $Λ$CDM in most cases, and only DESY5yr is preferred under both criteria. These findings show that the Hu-Sawicki $f(R)$ scenario with unconstrained curvature is nonetheless a statistically feasible, if not obviously preferred, alternative to $Λ$CDM, and that curvature restrictions generated inside $Λ$CDM cannot be viewed as independent of the underlying gravity model.",
      "title": "Hu-Sawicki $f(R)$ Gravity in a Non-Flat Universe: Constraints from DESI-DR2, BBN, and Type Ia Supernovae",
      "updated": "2026-09-15",
      "url": "https://arxiv.org/abs/2609.17643"
    },
    {
      "age_days": 2,
      "arxiv_id": "2609.16603",
      "authors": [
        "Jeng Wen Joshua Lean",
        "Ting-Yu Yen",
        "Wei-Fang Sun",
        "Simon See",
        "Hung-Kuo Chu",
        "Shih-Hsuan Hung"
      ],
      "content_date": "2026-09-15",
      "freshness": "fresh",
      "id": "arxiv:2609.16603",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-15",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Full-context neural visual geometry is impractical for thousands of images, while sequence-based chunking poorly captures irregular non-local overlap in multi-sequence aerial collections. We present Graph-Guided Neural Visual Geometry for Aerial Registration (G3AR), a graph-guided framework for scalable dense neural geometry. Before local inference, G3AR builds a geometrically verified image-proximity graph that guides bounded overlapping chunks and induces a chunk graph whose maximum spanning tree defines alignment topology. Compatible backbones process chunks independently; shared-image predictions then estimate three-dimensional similarity (Sim(3)) transforms that register local cameras and geometry in a common frame. Across four real aerial scenes, G3AR improves pose error and runtime in matched VGGT- and Pi3-backed comparisons, while its DA3 variant achieves the lowest pose error among evaluated neural-geometry methods.",
      "title": "G3AR: Graph-Guided Neural Visual Geometry for Scalable Multi-Sequence Aerial Registration",
      "updated": "2026-09-15",
      "url": "https://arxiv.org/abs/2609.16603"
    },
    {
      "age_days": 2,
      "arxiv_id": "2609.16470",
      "authors": [
        "Levent Alpöge",
        "Tristan Buckmaster",
        "Matei P. Coiculescu"
      ],
      "content_date": "2026-09-15",
      "freshness": "fresh",
      "id": "arxiv:2609.16470",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-15",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "By adapting the techniques used in the IPM blowup result of Córdoba-Martínez-Zoroa with spatially smooth force, we prove finite-time blow-up for the IPM equation on $\\mathbb T^2$ with a uniformly spacetime smooth force. In particular, we show there exist a smooth odd initial density, a smooth odd force $F\\in C^\\infty([0,1]\\times\\mathbb T^2)$, and a classical solution $ρ$ on $[0,1)$ whose density gradient and spatial velocity gradient diverge in $L^\\infty$ as $t\\uparrow 1$. Nevertheless, $ρ(t)$ converges in $C^η$ for every $0\\leqη<1$.",
      "title": "Extending the Córdoba-Martínez-Zoroa IPM Blow-Up to Uniformly Space-Time Smooth Forcing",
      "updated": "2026-09-15",
      "url": "https://arxiv.org/abs/2609.16470"
    },
    {
      "age_days": 2,
      "arxiv_id": "2609.16555",
      "authors": [
        "Nikita Lebedev"
      ],
      "content_date": "2026-09-15",
      "freshness": "fresh",
      "id": "arxiv:2609.16555",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-15",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "We study the augmented Zarankiewicz problem, in which disjoint pairs of cells are added to a binary matrix with no all-one $2\\times2$ submatrix. The pairs must satisfy compatibility conditions, and the objective counts each original occupied cell and each added pair once. We show that starting with a maximum $C_4$-free matrix can lower the final optimum, answering a question of Qi, Cui, and Xu. Let ${z_A}(m,n)$ be the optimum over all $C_4$-free initial matrices, and ${z_L}(m,n)$ the optimum when the initial matrix must have the maximum number of occupied cells. As $n\\to\\infty$ with $n\\le m=o(n^2)$, we prove \\[ {z_A}(m,n)-{z_L}(m,n)\\ge\\left(\\frac1{30}-o(1)\\right)mn \\] and determine the sharp second-order term: \\[ {z_A}(m,n)=\\frac{mn}{3}+\\left(\\frac1{\\sqrt6}+o(1)\\right)n\\sqrt m. \\] An explicit construction gives a separation at $m=n=1893$. We also find a sharp density threshold: when $n\\to\\infty$ and $m/n^2\\to c>0$, the limited density ${z_L}(m,n)/(mn)$ tends to $1/3$ if and only if $c\\ge1/12$. The proofs combine density and stability estimates, combinatorial constructions, and an exact polynomial certificate.",
      "title": "Density and separation for augmented Zarankiewicz numbers",
      "updated": "2026-09-15",
      "url": "https://arxiv.org/abs/2609.16555"
    },
    {
      "age_days": 3,
      "arxiv_id": "2609.15648",
      "authors": [
        "Son Ho",
        "Cédric Fournet",
        "Jonathan Protzenko",
        "Michael Naehrig",
        "Joshua Clune",
        "Patrick Longa",
        "Guillaume Boisseau",
        "Fernando Leal Sánchez",
        "Aymeric Fromherz",
        "Antoine Delignat-Lavaud"
      ],
      "content_date": "2026-09-14",
      "freshness": "fresh",
      "id": "arxiv:2609.15648",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-14",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "We develop a new methodology for verifying cryptographic software. We target production code written in Rust for performance and system integration, rather than verification convenience. Rust's ownership discipline enables Aeneas to extract a pure model of this code in Lean, relieving us from low-level reasoning about pointer liveness and aliasing. Lean's extensibility lets us develop tactics and libraries that greatly simplify reasoning about extracted Rust code. We design and tune our toolchain to facilitate the use of AI. Agents autonomously write formal proofs, which are independently verified by the Lean kernel. Agents also assist in the formalization of cryptographic standards and platform-specific intrinsics, which still requires expert design and review. We apply our methodology to SymCrypt, Microsoft's cryptographic provider. We verify its implementations of algorithms such as SHA-3 and ML-KEM, which were ported from C to Rust. We also extend SymCrypt with experimental optimizations and implementations of algorithms such as FrodoKEM, ML-DSA, and HPKE to explore the scalability of writing, adapting, and verifying cryptographic code. Our 237~KLOC Lean development establishes safety, panic-freedom, and functional correctness of 16.7~KLOC of Rust code supporting post-quantum cipher suites for x86-64 and ARM platforms. Our evaluation shows that verified Rust can meet SymCrypt's performance, portability, deployment, and maintainability requirements.",
      "title": "Scaling Verification of Cryptographic Software with Aeneas, Rust, and Lean",
      "updated": "2026-09-14",
      "url": "https://arxiv.org/abs/2609.15648"
    },
    {
      "age_days": 3,
      "arxiv_id": "2609.16299",
      "authors": [
        "Yuquan Fu",
        "Carlo Angiuli",
        "Sam Tobin-Hochstadt"
      ],
      "content_date": "2026-09-14",
      "freshness": "fresh",
      "id": "arxiv:2609.16299",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-14",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Over the past two decades, numerous systems have brought some of the benefits of dependent typing to a wide variety of new programming languages, often by restricting which terms can appear inside types. Such techniques are known as refinement types, occurrence typing, liquid types, and path dependent types, among others. However, the restrictions adopted by these systems often break the substitution property, because they explicitly disallow the ability to substitute arbitrary terms for variables inside types. This leads to significant complexity in the design and metatheory of these systems, increasing the possibility of significant errors. We consider a specific line of work on occurrence typing, namely, the calculus underlying Typed Racket due to Tobin-Hochstadt and Felleisen 2010. We show that the fundamental challenge of substitution into types resulted in multiple flaws in the formalism and the syntactic type soundness theorem of this work. These flaws are replicated in several other papers building on this work, and also surface as a soundness bug in Typed Racket itself. We identify and repair these problems, revising the core calculus of Typed Racket and giving a \\emph{semantic type soundness} proof using step-indexed logical relations, formalized in Lean. We argue that this approach is simpler than it may seem, and easily scales to handle the complexity of the occurrence typing in Typed Racket.",
      "title": "Revisiting Soundness for Occurrence Typing, Semantically",
      "updated": "2026-09-14",
      "url": "https://arxiv.org/abs/2609.16299"
    },
    {
      "age_days": 3,
      "arxiv_id": "2609.14925",
      "authors": [
        "Nathan Peterson",
        "Avik Mahata",
        "Nick Beaver",
        "Mohsen Kivy"
      ],
      "content_date": "2026-09-14",
      "freshness": "fresh",
      "id": "arxiv:2609.14925",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-14",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Refractory alloys with a ductile body-centered-cubic (BCC) matrix strengthened by ordered B2 precipitates offer a high-temperature analogue to the gamma/gamma-prime architecture of Ni-based superalloys. Ruthenium is particularly attractive as a B2 stabilizer because RuHf, RuZr, and RuTi can retain ordered phases well above 1300 C. In this work, equilibrium CALPHAD calculations were coupled with random-forest-guided active learning to explore a ten-element Nb-based, Ru-bearing composition space containing Nb, Ta, Mo, V, Ru, Ti, Zr, Hf, Al, and Y at 1 at.% resolution. Across 500 CALPHAD-evaluated alloys, the calculations reproduced the principal trends reported for the Ru-B2 design space. RuHf and RuZr remained stable to the solidus, RuTi commonly exhibited a solutionizing window, and Al-containing alloys preferentially formed competing sigma and A15 phases. The upper bound of the BCC+B2 field increased from a median of approximately 1570 C at 5 at.% Ru to approximately 1980 C near 9-10 at.% Ru. Among the group-IV additions, Hf, Zr, and Ti produced progressively lower two-phase stability. Re-screening using physically motivated criteria identified 100 alloys satisfying requirements for high-temperature BCC+B2 stability, absence of liquid, phase purity, and appropriate secondary-phase fraction, including 19 Ru-lean compositions and two independently reported HfRu-B2 alloys. The results establish practical compositional design rules for Ru-stabilized dual-phase refractory alloys and identify phase-specific BCC/B2 lattice misfit as a key target for future design cycles.",
      "title": "Machine Learning Guided CALPHAD Design of Ru-Stabilized BCC B2 Refractory Alloys",
      "updated": "2026-09-14",
      "url": "https://arxiv.org/abs/2609.14925"
    },
    {
      "age_days": 3,
      "arxiv_id": "2609.15554",
      "authors": [
        "Javier Aguilar Martín"
      ],
      "content_date": "2026-09-14",
      "freshness": "fresh",
      "id": "arxiv:2609.15554",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-14",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "We study packings of annuli of a common width, allowing each ring to nest inside the hole of a larger one. The objectives of maximizing contact area and cardinality diverge: area is superadditive in the radius, cardinality is not. Under superincreasing radii, every descending greedy maximizes every positive, strictly increasing, superadditive objective. More strongly, any choice among feasible containers yields the lexicographically maximal feasible set, for containers of arbitrary shape in every dimension. This placement irrelevance holds unconditionally for at most three rings and fails at four in disks and squares; twin instances exclude every universal rule based only on the observable state. Write $ρ=\\max_i(\\sum_{j>i}r_j)/r_i$. The additive model has threshold exactly $1$. For disks we prove the exact global threshold $τ=\\varphi$, with no failure at $ρ\\le\\varphi$, for every finite inventory, even with independent hole radii. The key geometric theorem states that, under golden tail bounds, an entire disk list fits a circular container if and only if its three largest disks fit; this supplies the uniform exchange of parents that the threshold proof needs. The Tribonacci constant $T\\approx1.83929$ remains the exact floor of a rigid subfamily. A dimension-reduction lemma transfers spherical sharpness results to all dimensions $d\\ge2$, and a separate argument proves the golden threshold for at most five rings in those dimensions. For square pans, a Cartesian confinement criterion gives twins and a family proving $τ_{\\square}\\le Y\\approx1.6845$; its optimality is open. For independent holes, the exact universal area guarantee under $ρ\\leκ<1$ is $\\min(1,κ^{-2}-1)$, with threshold $1/\\sqrt2$. The repository has 122 Lean theorems. Euclidean geometry, forest assembly and continuity remain written proofs; numerical checks do not substitute for them.",
      "title": "Greedy Packing of Nested Rings: Placement Rules, a Golden Counterexample, and a Tribonacci Floor",
      "updated": "2026-09-15",
      "url": "https://arxiv.org/abs/2609.15554"
    },
    {
      "age_days": 3,
      "arxiv_id": "2609.15986",
      "authors": [
        "Tzu-Chen Huang"
      ],
      "content_date": "2026-09-14",
      "freshness": "fresh",
      "id": "arxiv:2609.15986",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-14",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "We construct a complex spherical fusion category with cyclic Haagerup-Izumi fusion rules for every odd n >= 3, and deduce pseudo-unitary existence. The proof has four parts: explicit real coefficients and their quadratic identities; a contour calculation for a range of cubic Fourier coefficients; algebraic completion of all cubics; and categorical reconstruction. Matrix inversion supplies reflection, and an extension of the dimension-field automorphism supplies the positive-dimensional category.",
      "title": "Cyclic Haagerup-Izumi fusion categories at every odd order",
      "updated": "2026-09-14",
      "url": "https://arxiv.org/abs/2609.15986"
    },
    {
      "age_days": 4,
      "arxiv_id": "2609.14314",
      "authors": [
        "S P Suresh"
      ],
      "content_date": "2026-09-13",
      "freshness": "fresh",
      "id": "arxiv:2609.14314",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-13",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "We present a new proof of weak normalization for intuitionistic natural deduction. The distinguishing features of this proof are that it works only with cuts rather than cut segments, provides explicit local rules for determining whether to contract a whole proof or reduce one of its subproofs, and in the latter case, which subproof to reduce. We also discuss a formalization of the entire proof in Lean, and present a deterministic algorithm for weak normalization.",
      "title": "Simplified proofs of Weak Normalization for propositional logic",
      "updated": "2026-09-13",
      "url": "https://arxiv.org/abs/2609.14314"
    },
    {
      "age_days": 4,
      "arxiv_id": "2609.14525",
      "authors": [
        "Zhipeng Lu",
        "Sichen Wang"
      ],
      "content_date": "2026-09-13",
      "freshness": "fresh",
      "id": "arxiv:2609.14525",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-13",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "We study how far a fixed Ramsey upper bound can be improved by descending through blue neighborhoods in one vertex set while keeping a second set fixed. A weighted inequality in the two set sizes determines when the descent can stop. For the source bound specified here, the infimum diagonal exponent over all finite derivations lies in $[1.305,\\,1.307]$. A finite derivation gives $R(k,k)\\le3.69507^k$ for all sufficiently large $k$; a concave polygon proves the lower bound for every finite depth. We also characterize the infimum as a greatest fixed point and show that every larger exponent has a finite derivation valid uniformly for nearby clique-size ratios.",
      "title": "Retained-Set Descent for Diagonal Ramsey Numbers",
      "updated": "2026-09-13",
      "url": "https://arxiv.org/abs/2609.14525"
    },
    {
      "age_days": 4,
      "arxiv_id": "2609.17600",
      "authors": [
        "Prashant Suresh Kamble"
      ],
      "content_date": "2026-09-13",
      "freshness": "fresh",
      "id": "arxiv:2609.17600",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [
        "negative:vision_world_models"
      ],
      "published": "2026-09-13",
      "score": -2.5,
      "source": "arxiv-ai4math-core",
      "summary": "Burning neat hydrogen in aircraft turbines eliminates carbon emissions, yet rapid reaction rates trigger nozzle flashback hazards and high nitrogen oxide emissions across flight throttles. This study investigates aerothermal holding mechanisms, flashback safety margins, and emission pathways in a dual-swirl combustor across equivalence ratios from 0.55 to 1.00. Three-dimensional simulations combine curvature-corrected shear-stress transport turbulence closure, dual-rate finite-rate and eddy-dissipation chemical kinetics, and discrete ordinates radiation, validated against experimental laser benchmarks using ASME grid standards. Advancing engine throttle triggers a topological flame transition from a faceplate-attached M-flame at lean idle to a lifted V-flame above equivalence ratio 0.895, while wall flashback safety indices consistently exceed 3.42. Nitric oxide emissions transition from water-chaperoned nitrous oxide intermediate reactions at lean idle (28.01 ppm, EINOx = 1.85 g/kg), scaling to 319.21 ppm (EINOx = 35.40 g/kg) at takeoff under thermal Zeldovich dominance, governed by a power-law exponent of 4.92. These findings deliver validated operability limits and establish an accessible workstation-based screening methodology for practical zero-carbon aero-engine combustor development.",
      "title": "Topological Flame Bifurcation, Aerodynamic Flashback Margins, and Multi-Pathway NOx Scaling in a 3D Swirl-Stabilized 100% Pure Hydrogen Aero-Engine Combustor",
      "updated": "2026-09-13",
      "url": "https://arxiv.org/abs/2609.17600"
    },
    {
      "age_days": 4,
      "arxiv_id": "2609.14572",
      "authors": [
        "Sushan Adhikari"
      ],
      "content_date": "2026-09-13",
      "freshness": "fresh",
      "id": "arxiv:2609.14572",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [
        "negative:generic_llm_rag"
      ],
      "published": "2026-09-13",
      "score": -2.5,
      "source": "arxiv-ai4math-core",
      "summary": "Teaching abstract theoretical computer science (TCS) concepts such as algorithm analysis and complexity theory is challenging because students must handle formal proofs and asymptotic reasoning that conventional resources rarely explain in an adaptive, on-demand way. We present AlgoRAG, a specialized Retrieval-Augmented Generation (RAG) system that couples a large language model (LLM) with a curated, domain-specific knowledge base to address these challenges. The knowledge base integrates authoritative textbooks, 847 lecture slides, 312 practice problems with solutions, 156 worked proof templates, and 89 complexity worksheets. AlgoRAG incorporates domain-specific optimizations including mathematical entity recognition, notation-aware retrieval, and pedagogical re-ranking. We evaluate AlgoRAG on 179 curated exam-style questions spanning asymptotic analysis, recurrence relations, dynamic programming, graph algorithms, NP-completeness, sorting, and divide-and-conquer. The system achieves a 100% success rate with a mean response time of 38.0 seconds. While BLEU-4 scores are zero -- a known limitation of n-gram matching on mathematical proofs where equivalent reasoning may use entirely different notation -- AlgoRAG attains ROUGE-1 F1 of 0.0963, ROUGE-L F1 of 0.0683, and a pedagogical quality score of 0.7620, indicating that responses are well-structured and didactically sound even when surface wording diverges from reference answers. Performance is especially strong on NP-completeness (ROUGE-1 F1 = 0.1285, pedagogical quality = 0.7643) and graph algorithms (ROUGE-1 F1 = 0.1023, pedagogical quality = 0.8086). These results support the conclusion that RAG is an effective architecture for personalized theoretical-CS instruction, providing correct, context-rich explanations even for highly abstract topics.",
      "title": "AlgoRAG: Retrieval-Augmented Generation for Theoretical Computer Science Education -- A Comprehensive Evaluation Framework for Algorithm Analysis and Complexity Theory",
      "updated": "2026-09-13",
      "url": "https://arxiv.org/abs/2609.14572"
    }
  ],
  "lookback_days": 21,
  "schema": "ai4math-radar-run-v1",
  "timezone": "America/Los_Angeles"
}